A brief summary of percolation theory

As mentioned before, the disordered solids will be mostly modelled in this book using randomly diluted site or bond lattice models. A knowledge of percolation cluster statistics will therefore be necessary and widely employed. Although this lattice percolation kind of disorder will not be the only kind of disorder used to model such solids, as can be seen later in this book, the widely established results for percolation statistics have been employed successsfully to understand and formulate analytically various breakdown properties of disordered solids. We therefore give here a very brief introduction to the percolation theory. For details, see the book by Stauffer and Aharony (1992). 

Similarly, one can study the growth of the elastic constants (say the rigidity modulus) of a randomly formed elastic network, near the percolation point. The central force elastic problem (for networks formed out of linear springs only) belongs however to a different class of percolation problem, known as elastic percolation or central force percolation, and is discussed separately later (see Section 1.2.1(f)). 

The growth of the conductivity or elasticity of such networks near their respective percolation threshold points can be expressed as powers (known as exponents) of the interval (of random concentration of the conducting or elastic material) from the percolation threshold. These powers or the exponents are observed to be universal, in the sense that they do not depend on many details of the problem or of the lattice, but depend on only some subtle geometric features of the problem e.g., the exponents often depend only on the lattice dimensionality. 

In order to make the discussion more quantitative and precise, let us 

We now define some statistical quantities of interest in percolation theory. Let ns p) denote the number of clusters (per lattice site) of size s. In fact, a detailed knowledge of ns p) would give us a lot of information on the percolation statistics, as most of the quantities of interest can be extracted from various moments of the cluster size distribution n. Although, in general, we do not have any analytic knowledge about this distribution function ns p) near Pc, we can utilise the powerful observation of scaling behaviour of ns p) near Pc (see the next section). 

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